 2.6  If \(x\in \mathfrak {m}\setminus \mathfrak {m}^2\) . Set \(N=M/xM\)  and \(K=\mathfrak {m}/(x)\)  then we have Singh's equality ( for \(M=A\)  see [1]}, and for the module case see [2]})
\(H(M,n)=\ell (N/K^{n+1}N)-\ell \left(\frac{\mathfrak {m}^{n+1}M:x}{\mathfrak {m}^nM}\right)\ \text{for all}\ n\ge 0.\) 
